516 BAGNOLD [chap. 21 



Avhere h is the constant flow depth, H is the energy head, and clHjdx is the 

 energy gradient, tan ^, due to the gravity slope. 



The A\hole power transmitted forward by a train of surface waves of height 

 H is given by l,pgH -Cn, where u is the proportion of energy travelling forward 

 with Mave phase velocity C. Hence the power co per unit of bed area is in the 

 present context the decrement of the transmitted power per unit of distance 

 travelled by the waves. Thus, oj = lpg d{H^Cn)ldx, which for propagation over 

 constant water depth and therefore at constant velocity, becomes 



which is of j)recisely the same form as relation (5). 



In arriving at (6) it is, of course, assumed that the decrement of the trans- 

 mitted power is due entirely to energy losses by fluid drag at the bed surface. 

 When the waves break, the much larger power decrement is likely to be due 

 mainly to other internal energy losses not associated with bed drag. This needs 

 fm-ther study, in particular of the variation in the amount of energy loss by bed 

 drag with variation in the kind of break which takes place. 



In the case of a ground wind, as in that of a sea-bed current, the fluid flow 

 relative to the boundary is not maintained by a gravity fall but by an in- 

 determinate combination of a j)ressure gradient, of the inertia of large fluid 

 masses, and of a downward transmission of shear stress from above. In such 

 cases the power oj may be inferred from the following general considerations 

 aj)plicable to all cases. 



The power, co, is essentially measureable as the product of the boundary 

 stress, ^F, arising directly or indirectly from the motion of the fluid, times its 

 flow velocity, u', at some effective distance from the bed boundary. An ex- 

 amination of the experimental data on wind-blown sand (Bagnold, 1936) 

 suggests the effective distance to be that of the "centre of pressure" at which 

 fluid momentum is transferred to the bed-load grain dispersion. 



In general, writing u' = CrU^=Cr-\/{-^ fJp), where Cr is a numerical coefficient 

 to be determined experimentally, we can define the available power as : 



OJ = Cr^F^^'^IVP = CrpU^. (7) 



In this fonU, by assuming Cr to be a general constant under conditions of no 

 suspended load, and by making certain quite general assumptions as to the 

 theoretical value of Creb, it was found j^ossible (Bagnold, 1956) to predict 

 the value of ii, to a good approximation from relation (4) both for water-driven 

 sands in laboratory flumes and for wind-driven sands. 



Moreover, the bed-load theory as given by Bagnold (1956) ajjpears to account 

 satisfactorily not only for the variation of ii, with grain size but also for the fact 

 that beds of grains smaUer than the order of 1 mm diameter become spon- 

 taneously rippled under the flow of water whereas beds of larger grains do not. 



